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A372559 a(n) is the index of the first occurrence of n in A371091.

%I A372559 #20 May 13 2024 09:15:04
%S A372559 0,1,3,9,21,51,111,321,741,2001,4311,8931,22791,52821,112881,293061,
%T A372559 803571,1824591,4887651,14587341,33986721,92184861,208581141,
%U A372559 431674011,877859751,2216416971,4893531411,11363224641,24302611101,63120770481,140757089241,341317579371,742438559631,1945801500411,4352527381971,11773265516781
%N A372559 a(n) is the index of the first occurrence of n in A371091.
%C A372559 The pattern in the primorial base expansion (A049345) of the terms is constructed recursively, so that the digit-positions of the primorial base expansion are successively filled with the positive terms of this sequence (1, 3, 9, 21, ...), up to that term that still fits to the position, i.e., is less than prime(i), for the positions i >= 1 indexed from the least significant end of the expansion. The nonleading digits are "frozen", and only the most significant digit keeps on increasing from a(1) to the maximal allowed a(x) for its position, after which the next term's expansion is obtained by prepending 1 to the front. See the examples.
%H A372559 Antti Karttunen, <a href="/A372559/b372559.txt">Table of n, a(n) for n = 0..1001</a>
%H A372559 <a href="/index/Pri#primorialbase">Index entries for sequences related to primorial base</a>
%F A372559 For n >= 0, A371091(a(n)) = n, and for all k < a(n), A371091(k) < n.
%e A372559    n,      a(n)     in primorial base
%e A372559    0,         0 =             0
%e A372559    1,         1 =             1
%e A372559    2,         3 =            11
%e A372559    3,         9 =           111
%e A372559    4,        21 =           311 (3 is less than prime(3)=5, so can be used now)
%e A372559    5,        51 =          1311 (9 cannot yet be used, so append 1 to the front)
%e A372559    6,       111 =          3311 (and then replace by next higher term that fits)
%e A372559    7,       321 =         13311
%e A372559    8,       741 =         33311
%e A372559    9,      2001 =         93311 (9 is less than prime(5)=11, so can be used now)
%e A372559   10,      4311 =        193311
%e A372559   11,      8931 =        393311
%e A372559   12,     22791 =        993311
%e A372559   13,     52821 =       1993311
%e A372559   14,    112881 =       3993311
%e A372559   15,    293061 =       9993311
%e A372559   16,    803571 =      19993311
%e A372559   17,   1824591 =      39993311
%e A372559   18,   4887651 =      99993311
%e A372559   19,  14587341 =     199993311
%e A372559   20,  33986721 =     399993311
%e A372559   21,  92184861 =     999993311
%e A372559   22, 208581141 =  {21}99993311 (21 is less than prime(9)=23, so can be used now)
%e A372559   23, 431674011 = 1{21}99993311
%e A372559 etc.
%o A372559 (PARI)
%o A372559 A002110(n) = prod(i=1,n,prime(i));
%o A372559 A235224(n) = { my(s=0, p=2); while(n, s++; n = n\p; p = nextprime(1+p)); (s); };
%o A372559 A276153(n) = { my(p=2,d=0); while(n, d = n%p; n = n\p; p = nextprime(1+p)); (d); };
%o A372559 memoA372559 = Map();
%o A372559 A372559(n) = if(n<=2, n+(n>1), my(v); if(mapisdefined(memoA372559,n,&v), v, my(prev=A372559(n-1), hi=A235224(prev), hd=A276153(prev),k=0,u); while(A372559(k)<hd, k++); u = A372559(1+k); v = if(u>=prime(hi), prev+A002110(hi), prev+((u-hd)*A002110(hi-1))); mapput(memoA372559,n,v); (v)));
%Y A372559 Positions of records in A371091.
%Y A372559 Cf. A002110, A235224, A276153.
%K A372559 nonn,base
%O A372559 0,3
%A A372559 _Antti Karttunen_, May 11 2024